A half-century ago, my professor derived these formulas for us.. . We have never had to factor-out-4 or factor-out-9.

[rant]

I've made this query at other math sites.

No one has ever thanked me for these streamlined formulas.

Instead, someone replies with something like:
"I'd rather have my students understand the concepts. . rather than memorizing a list of formulas"
as if memorizing is a Bad Thing.

My first response has been:
If you've already memorized a formula with uČ + 1. . why not expand it to include the "a"?

They usually reply with hints that I'm a formula-happy moron.
(Have they ever seen any of my posts?
Isn't Understanding the main feature of all my explanations?)

And then I must respond with something like:
If Understanding is the most important issue,. . then we must not memorize the Quadratic Formula.
Instead, we make the student complete-the-square every time.

Then someone says, "What's wrong with completing-the-square?
I'd rather have them understand the procedure, than just memorize a formula."

Then I must explain the Philosophy of Learning.
When we learn a new fact, we in fact memorize it.. . Yes, we "commit it to memory" for future use.
Then (and here's the important part) . . . we move on.
We add the formula/shortcut to our arsenal (in the war against our own ignorance). . and progress to bigger and better things.
And it is hoped that we retain the reasoning behind the formulas/shortcuts.

We do not draw a line in the sand and say, "This is far enough!. . From now on, everything will be derived ... no more blind memorization!"

If Memorization is a misdemeanor, then 2 Ś 3 .= .6 is at least a felony.. . After all, it can derived from: 3 + 3 .= .6